3.1.40 \(\int x^3 (a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2)) \, dx\) [40]

Optimal. Leaf size=591 \[ -\frac {45 b^3 n^3 x^2}{128 d f}+\frac {3}{64} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2} \]

[Out]

-45/128*b^3*n^3*x^2/d/f+3/64*b^3*n^3*x^4+21/32*b^2*n^2*x^2*(a+b*ln(c*x^n))/d/f-9/64*b^2*n^2*x^4*(a+b*ln(c*x^n)
)-9/16*b*n*x^2*(a+b*ln(c*x^n))^2/d/f+3/16*b*n*x^4*(a+b*ln(c*x^n))^2+1/4*x^2*(a+b*ln(c*x^n))^3/d/f-1/8*x^4*(a+b
*ln(c*x^n))^3+3/128*b^3*n^3*ln(d*f*x^2+1)/d^2/f^2-3/128*b^3*n^3*x^4*ln(d*f*x^2+1)-3/32*b^2*n^2*(a+b*ln(c*x^n))
*ln(d*f*x^2+1)/d^2/f^2+3/32*b^2*n^2*x^4*(a+b*ln(c*x^n))*ln(d*f*x^2+1)+3/16*b*n*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)
/d^2/f^2-3/16*b*n*x^4*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)-1/4*(a+b*ln(c*x^n))^3*ln(d*f*x^2+1)/d^2/f^2+1/4*x^4*(a+b
*ln(c*x^n))^3*ln(d*f*x^2+1)-3/64*b^3*n^3*polylog(2,-d*f*x^2)/d^2/f^2+3/16*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-d
*f*x^2)/d^2/f^2-3/8*b*n*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^2)/d^2/f^2-3/32*b^3*n^3*polylog(3,-d*f*x^2)/d^2/f^2
+3/8*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^2)/d^2/f^2-3/16*b^3*n^3*polylog(4,-d*f*x^2)/d^2/f^2

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2504, 2442, 45, 2424, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \frac {3 b^2 n^2 \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{16 d^2 f^2}+\frac {3 b^2 n^2 \text {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}-\frac {3 b n \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (2,-d f x^2\right )}{64 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (3,-d f x^2\right )}{32 d^2 f^2}-\frac {3 b^3 n^3 \text {PolyLog}\left (4,-d f x^2\right )}{16 d^2 f^2}-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{32 d^2 f^2}+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}+\frac {3}{32} b^2 n^2 x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{16 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{16} b n x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{128 d^2 f^2}-\frac {45 b^3 n^3 x^2}{128 d f}-\frac {3}{128} b^3 n^3 x^4 \log \left (d f x^2+1\right )+\frac {3}{64} b^3 n^3 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(-45*b^3*n^3*x^2)/(128*d*f) + (3*b^3*n^3*x^4)/64 + (21*b^2*n^2*x^2*(a + b*Log[c*x^n]))/(32*d*f) - (9*b^2*n^2*x
^4*(a + b*Log[c*x^n]))/64 - (9*b*n*x^2*(a + b*Log[c*x^n])^2)/(16*d*f) + (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 +
(x^2*(a + b*Log[c*x^n])^3)/(4*d*f) - (x^4*(a + b*Log[c*x^n])^3)/8 + (3*b^3*n^3*Log[1 + d*f*x^2])/(128*d^2*f^2)
- (3*b^3*n^3*x^4*Log[1 + d*f*x^2])/128 - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/(32*d^2*f^2) + (3*b^
2*n^2*x^4*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/32 + (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(16*d^2*f^2)
- (3*b*n*x^4*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/16 - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(4*d^2*f^2)
+ (x^4*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/4 - (3*b^3*n^3*PolyLog[2, -(d*f*x^2)])/(64*d^2*f^2) + (3*b^2*n^2
*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*x^2)])/(16*d^2*f^2) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*x^2)])
/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[3, -(d*f*x^2)])/(32*d^2*f^2) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*
f*x^2)])/(8*d^2*f^2) - (3*b^3*n^3*PolyLog[4, -(d*f*x^2)])/(16*d^2*f^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2423

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2424

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-(3 b n) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d f}-\frac {1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {1}{8} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right ) \, dx+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{x} \, dx}{4 d^2 f^2}-\frac {(3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d f}\\ &=-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )\right ) \, dx+\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{x} \, dx}{4 d^2 f^2}+\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d f}\\ &=-\frac {3 b^3 n^3 x^2}{16 d f}+\frac {3}{256} b^3 n^3 x^4+\frac {3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{8 d f}-\frac {3}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{16} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right ) \, dx-\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x} \, dx}{8 d^2 f^2}+\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 d f}-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f x^2\right )}{x} \, dx}{8 d^2 f^2}\\ &=-\frac {9 b^3 n^3 x^2}{32 d f}+\frac {3}{128} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{8} \left (3 b^3 n^3\right ) \int \left (\frac {x}{4 d f}-\frac {x^3}{8}-\frac {\log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \log \left (1+d f x^2\right )\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f x^2\right )}{x} \, dx}{16 d^2 f^2}\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{32} \left (3 b^3 n^3\right ) \int x^3 \log \left (1+d f x^2\right ) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f x^2\right )}{x} \, dx}{32 d^2 f^2}\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}-\frac {1}{64} \left (3 b^3 n^3\right ) \text {Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac {1}{128} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {x^2}{1+d f x} \, dx,x,x^2\right )\\ &=-\frac {21 b^3 n^3 x^2}{64 d f}+\frac {9}{256} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}+\frac {1}{128} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \left (-\frac {1}{d^2 f^2}+\frac {x}{d f}+\frac {1}{d^2 f^2 (1+d f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {45 b^3 n^3 x^2}{128 d f}+\frac {3}{64} b^3 n^3 x^4+\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{32 d f}-\frac {9}{64} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{16 d f}+\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{128 d^2 f^2}-\frac {3}{128} b^3 n^3 x^4 \log \left (1+d f x^2\right )-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{32 d^2 f^2}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \text {Li}_2\left (-d f x^2\right )}{64 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{16 d^2 f^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_3\left (-d f x^2\right )}{32 d^2 f^2}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )}{8 d^2 f^2}-\frac {3 b^3 n^3 \text {Li}_4\left (-d f x^2\right )}{16 d^2 f^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.60, size = 1234, normalized size = 2.09 \begin {gather*} -\frac {-2 d f x^2 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+d^2 f^2 x^4 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-2 d^2 f^2 x^4 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+12 b \left (8 a^2-4 a b n+b^2 n^2\right ) \log \left (c x^n\right )-24 b^2 (-4 a+b n) \log ^2\left (c x^n\right )+32 b^3 \log ^3\left (c x^n\right )\right ) \log \left (1+d f x^2\right )+2 \left (32 a^3-24 a^2 b n+12 a b^2 n^2-3 b^3 n^3+48 a b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+96 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+96 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2-24 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+32 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right ) \log \left (1+d f x^2\right )+24 b n \left (8 a^2-4 a b n+b^2 n^2+4 b^2 n \left (n \log (x)-\log \left (c x^n\right )\right )+16 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+8 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\frac {1}{2} d f x^2-\frac {1}{8} d^2 f^2 x^4-d f x^2 \log (x)+\frac {1}{2} d^2 f^2 x^4 \log (x)+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )-96 b^2 n^2 \left (4 a-b n-4 b n \log (x)+4 b \log \left (c x^n\right )\right ) \left (\frac {1}{4} d f x^2 \left (1-2 \log (x)+2 \log ^2(x)\right )-\frac {1}{32} d^2 f^2 x^4 \left (1-4 \log (x)+8 \log ^2(x)\right )-\frac {1}{2} \log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )-\frac {1}{2} \log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-\log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )+b^3 n^3 \left (-16 d f x^2 \left (-3+6 \log (x)-6 \log ^2(x)+4 \log ^3(x)\right )+d^2 f^2 x^4 \left (-3+12 \log (x)-24 \log ^2(x)+32 \log ^3(x)\right )+64 \left (\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )\right )+64 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right )}{256 d^2 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)],x]

[Out]

-1/256*(-2*d*f*x^2*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a
^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^
2 - 24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + d^2*f^2*x^4*(32*a^3 - 24*a^
2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*b*(-(n*Log[x]) + Log[c*x^n]) +
12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 24*b^3*n*(-(n*Log[x]) + Log[c*
x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 2*d^2*f^2*x^4*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3
+ 12*b*(8*a^2 - 4*a*b*n + b^2*n^2)*Log[c*x^n] - 24*b^2*(-4*a + b*n)*Log[c*x^n]^2 + 32*b^3*Log[c*x^n]^3)*Log[1
+ d*f*x^2] + 2*(32*a^3 - 24*a^2*b*n + 12*a*b^2*n^2 - 3*b^3*n^3 + 48*a*b^2*n*(n*Log[x] - Log[c*x^n]) + 96*a^2*
b*(-(n*Log[x]) + Log[c*x^n]) + 12*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 96*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 -
24*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 32*b^3*(-(n*Log[x]) + Log[c*x^n])^3)*Log[1 + d*f*x^2] + 24*b*n*(8*a^2
- 4*a*b*n + b^2*n^2 + 4*b^2*n*(n*Log[x] - Log[c*x^n]) + 16*a*b*(-(n*Log[x]) + Log[c*x^n]) + 8*b^2*(-(n*Log[x]
) + Log[c*x^n])^2)*((d*f*x^2)/2 - (d^2*f^2*x^4)/8 - d*f*x^2*Log[x] + (d^2*f^2*x^4*Log[x])/2 + Log[x]*Log[1 - I
*Sqrt[d]*Sqrt[f]*x] + Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*
Sqrt[d]*Sqrt[f]*x]) - 96*b^2*n^2*(4*a - b*n - 4*b*n*Log[x] + 4*b*Log[c*x^n])*((d*f*x^2*(1 - 2*Log[x] + 2*Log[x
]^2))/4 - (d^2*f^2*x^4*(1 - 4*Log[x] + 8*Log[x]^2))/32 - (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x])/2 - (Log[x]^2
*Log[1 + I*Sqrt[d]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - Log[x]*PolyLog[2, I*Sqrt[d]*Sqr
t[f]*x] + PolyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x] + PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + b^3*n^3*(-16*d*f*x^2*(-3 + 6
*Log[x] - 6*Log[x]^2 + 4*Log[x]^3) + d^2*f^2*x^4*(-3 + 12*Log[x] - 24*Log[x]^2 + 32*Log[x]^3) + 64*(Log[x]^3*L
og[1 + I*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, (-I)*Sqrt[d]
*Sqrt[f]*x] + 6*PolyLog[4, (-I)*Sqrt[d]*Sqrt[f]*x]) + 64*(Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 3*Log[x]^2*P
olyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 6*Log[x]*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x] + 6*PolyLog[4, I*Sqrt[d]*Sqrt[f]*x])
))/(d^2*f^2)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)

[Out]

int(x^3*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="maxima")

[Out]

1/128*(32*b^3*x^4*log(x^n)^3 - 24*(b^3*(n - 4*log(c)) - 4*a*b^2)*x^4*log(x^n)^2 + 12*((n^2 - 4*n*log(c) + 8*lo
g(c)^2)*b^3 - 4*a*b^2*(n - 4*log(c)) + 8*a^2*b)*x^4*log(x^n) + (12*(n^2 - 4*n*log(c) + 8*log(c)^2)*a*b^2 - (3*
n^3 - 12*n^2*log(c) + 24*n*log(c)^2 - 32*log(c)^3)*b^3 - 24*a^2*b*(n - 4*log(c)) + 32*a^3)*x^4)*log(d*f*x^2 +
1) - integrate(1/64*(32*b^3*d*f*x^5*log(x^n)^3 + 24*(4*a*b^2*d*f - (d*f*n - 4*d*f*log(c))*b^3)*x^5*log(x^n)^2
+ 12*(8*a^2*b*d*f - 4*(d*f*n - 4*d*f*log(c))*a*b^2 + (d*f*n^2 - 4*d*f*n*log(c) + 8*d*f*log(c)^2)*b^3)*x^5*log(
x^n) + (32*a^3*d*f - 24*(d*f*n - 4*d*f*log(c))*a^2*b + 12*(d*f*n^2 - 4*d*f*n*log(c) + 8*d*f*log(c)^2)*a*b^2 -
(3*d*f*n^3 - 12*d*f*n^2*log(c) + 24*d*f*n*log(c)^2 - 32*d*f*log(c)^3)*b^3)*x^5)/(d*f*x^2 + 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="fricas")

[Out]

integral(b^3*x^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*x^3*log(d*f*x^2 + 1)*log(c*x^n)^2 + 3*a^2*b*x^3*log(d
*f*x^2 + 1)*log(c*x^n) + a^3*x^3*log(d*f*x^2 + 1), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2)),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))^3*log(d*(1/d+f*x^2)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3,x)

[Out]

int(x^3*log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3, x)

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